Thomas' Calculus Early Transcendentals 1292021233, 9781292021232

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Thomas' Calculus Early Transcendentals George B. Thomas Jr. Maurice D. Weir Joel R. Hass Twelfth Edition

Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners.

ISBN 10: 1-292-02123-3 ISBN 13: 978-1-292-02123-2

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

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Table of Contents Chapter 1. Functions George B. Thomas Jr./Maurice D. Weir/Joel Hass

1

Chapter 2. Limits and Continuity George B. Thomas Jr./Maurice D. Weir/Joel Hass

58

Chapter 3. Differentiation George B. Thomas Jr./Maurice D. Weir/Joel Hass

122

Chapter 4. Applications of Derivatives George B. Thomas Jr./Maurice D. Weir/Joel Hass

222

Chapter 5. Integration George B. Thomas Jr./Maurice D. Weir/Joel Hass

297

Chapter 6. Applications of Definite Integrals George B. Thomas Jr./Maurice D. Weir/Joel Hass

363

Chapter 7. Integrals and Transcendental Functions George B. Thomas Jr./Maurice D. Weir/Joel Hass

417

Chapter 8. Techniques of Integration George B. Thomas Jr./Maurice D. Weir/Joel Hass

453

Chapter 9. First-Order Differential Equations George B. Thomas Jr./Maurice D. Weir/Joel Hass

514

Chapter 10. Infinite Sequences and Series George B. Thomas Jr./Maurice D. Weir/Joel Hass

550

Chapter 11. Parametric Equations and Polar Coordinates George B. Thomas Jr./Maurice D. Weir/Joel Hass

628

Chapter 12. Vectors and the Geometry of Space George B. Thomas Jr./Maurice D. Weir/Joel Hass

678

Chapter 13. Vector-Valued Functions and Motion in Space George B. Thomas Jr./Maurice D. Weir/Joel Hass

725

Chapter 14. Partial Derivatives George B. Thomas Jr./Maurice D. Weir/Joel Hass

765

I

II

Chapter 15. Multiple Integrals George B. Thomas Jr./Maurice D. Weir/Joel Hass

854

Chapter 16. Integration in Vector Fields George B. Thomas Jr./Maurice D. Weir/Joel Hass

919

Chapter 17. Second-Order Differential Equations George B. Thomas Jr./Maurice D. Weir/Joel Hass

1007

Answers to Odd-Numbered Exercises George B. Thomas Jr./Maurice D. Weir/Joel Hass

1041

Credits George B. Thomas Jr./Maurice D. Weir/Joel Hass

1113

Index

1115

FPO

1 FUNCTIONS OVERVIEW Functions are fundamental to the study of calculus. In this chapter we review what functions are and how they are pictured as graphs, how they are combined and transformed, and ways they can be classified. We review the trigonometric functions, and we discuss misrepresentations that can occur when using calculators and computers to obtain a function’s graph. We also discuss inverse, exponential, and logarithmic functions. The real number system, Cartesian coordinates, straight lines, parabolas, and circles are reviewed in the Appendices.

1.1

Functions and Their Graphs Functions are a tool for describing the real world in mathematical terms. A function can be represented by an equation, a graph, a numerical table, or a verbal description; we will use all four representations throughout this book. This section reviews these function ideas.

Functions; Domain and Range The temperature at which water boils depends on the elevation above sea level (the boiling point drops as you ascend). The interest paid on a cash investment depends on the length of time the investment is held. The area of a circle depends on the radius of the circle. The distance an object travels at constant speed along a straight-line path depends on the elapsed time. In each case, the value of one variable quantity, say y, depends on the value of another variable quantity, which we might call x. We say that “y is a function of x” and write this symbolically as y = ƒ(x)

(“y equals ƒ of x”).

In this notation, the symbol ƒ represents the function, the letter x is the independent variable representing the input value of ƒ, and y is the dependent variable or output value of ƒ at x.

DEFINITION A function ƒ from a set D to a set Y is a rule that assigns a unique (single) element ƒsxd H Y to each element x H D. The set D of all possible input values is called the domain of the function. The set of all values of ƒ(x) as x varies throughout D is called the range of the function. The range may not include every element in the set Y. The domain and range of a function can be any sets of objects, but often in calculus they are sets of real numbers interpreted as points of a coordinate line. (In Chapters 13–16, we will encounter functions for which the elements of the sets are points in the coordinate plane or in space.) From Chapter 1 of Thomas' Calculus Early Transcendentals, Twelfth Edition. George B. Thomas Jr., Maurice D. Weir, Joel Hass. Copyright © 2010 by Pearson Education, Inc. All rights reserved.

1

2

x

Chapter 1: Functions

f

Input (domain)

Output (range)

f (x)

FIGURE 1.1 A diagram showing a function as a kind of machine.

x a D  domain set

f(a)

f (x)

Y  set containing the range

FIGURE 1.2 A function from a set D to a set Y assigns a unique element of Y to each element in D.

Often a function is given by a formula that describes how to calculate the output value from the input variable. For instance, the equation A = pr 2 is a rule that calculates the area A of a circle from its radius r (so r, interpreted as a length, can only be positive in this formula). When we define a function y = ƒsxd with a formula and the domain is not stated explicitly or restricted by context, the domain is assumed to be the largest set of real x-values for which the formula gives real y-values, the so-called natural domain. If we want to restrict the domain in some way, we must say so. The domain of y = x 2 is the entire set of real numbers. To restrict the domain of the function to, say, positive values of x, we would write “y = x 2, x 7 0.” Changing the domain to which we apply a formula usually changes the range as well. The range of y = x 2 is [0, q d. The range of y = x 2, x Ú 2, is the set of all numbers obtained by squaring numbers greater than or equal to 2. In set notation (see Appendix 1), the range is 5x 2 ƒ x Ú 26 or 5y ƒ y Ú 46 or [4, q d. When the range of a function is a set of real numbers, the function is said to be realvalued. The domains and ranges of many real-valued functions of a real variable are intervals or combinations of intervals. The intervals may be open, closed, or half open, and may be finite or infinite. The range of a function is not always easy to find. A function ƒ is like a machine that produces an output value ƒ(x) in its range whenever we feed it an input value x from its domain (Figure 1.1). The function keys on a calculator give an example of a function as a machine. For instance, the 2x key on a calculator gives an output value (the square root) whenever you enter a nonnegative number x and press the 2x key. A function can also be pictured as an arrow diagram (Figure 1.2). Each arrow associates an element of the domain D with a unique or single element in the set Y. In Figure 1.2, the arrows indicate that ƒ(a) is associated with a, ƒ(x) is associated with x, and so on. Notice that a function can have the same value at two different input elements in the domain (as occurs with ƒ(a) in Figure 1.2), but each input element x is assigned a single output value ƒ(x).

EXAMPLE 1

Let’s verify the natural domains and associated ranges of some simple functions. The domains in each case are the values of x for which the formula makes sense.

Function y y y y y

= = = = =

x2 1>x 2x 24 - x 21 - x 2

Domain (x)

Range ( y)

s - q, q d s - q , 0d ´ s0, q d [0, q d s - q , 4] [-1, 1]

[0, q d s - q , 0d ´ s0, q d [0, q d [0, q d [0, 1]

The formula y = x 2 gives a real y-value for any real number x, so the domain is s - q , q d. The range of y = x 2 is [0, q d because the square of any real number is nonnegative and every nonnegative number y is the square of its own square root, y = A 2y B 2 for y Ú 0. The formula y = 1>x gives a real y-value for every x except x = 0. For consistency in the rules of arithmetic, we cannot divide any number by zero. The range of y = 1>x, the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since y = 1>(1>y). That is, for y Z 0 the number x = 1>y is the input assigned to the output value y. The formula y = 1x gives a real y-value only if x Ú 0. The range of y = 1x is [0, q d because every nonnegative number is some number’s square root (namely, it is the square root of its own square). In y = 14 - x, the quantity 4 - x cannot be negative. That is, 4 - x Ú 0, or x … 4. The formula gives real y-values for all x … 4. The range of 14 - x is [0, q d, the set of all nonnegative numbers. Solution

2

3

1.1 Functions and Their Graphs

The formula y = 21 - x 2 gives a real y-value for every x in the closed interval from - 1 to 1. Outside this domain, 1 - x 2 is negative and its square root is not a real number. The values of 1 - x 2 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of 21 - x 2 is [0, 1].

Graphs of Functions If ƒ is a function with domain D, its graph consists of the points in the Cartesian plane whose coordinates are the input-output pairs for ƒ. In set notation, the graph is 5sx, ƒsxdd ƒ x H D6. The graph of the function ƒsxd = x + 2 is the set of points with coordinates (x, y) for which y = x + 2. Its graph is the straight line sketched in Figure 1.3. The graph of a function ƒ is a useful picture of its behavior. If (x, y) is a point on the graph, then y = ƒsxd is the height of the graph above the point x. The height may be positive or negative, depending on the sign of ƒsxd (Figure 1.4). y

f(1)

y

f (2) x

yx2

0

1

x

2 f(x)

2 (x, y) –2

x

y  x2

-2 -1 0 1

4 1 0 1

3 2

9 4

2

4

x

0

FIGURE 1.4 If (x, y) lies on the graph of ƒ, then the value y = ƒsxd is the height of the graph above the point x (or below x if ƒ(x) is negative).

FIGURE 1.3 The graph of ƒsxd = x + 2 is the set of points (x, y) for which y has the value x + 2 .

EXAMPLE 2

Graph the function y = x 2 over the interval [-2, 2].

Make a table of xy-pairs that satisfy the equation y = x 2. Plot the points (x, y) whose coordinates appear in the table, and draw a smooth curve (labeled with its equation) through the plotted points (see Figure 1.5).

Solution

y

How do we know that the graph of y = x 2 doesn’t look like one of these curves? (–2, 4)

(2, 4)

4

y

y

y  x2 3 ⎛3 , 9⎛ ⎝2 4⎝

2 (–1, 1)

1

–2

0

–1

1

2

y  x 2?

y  x 2?

(1, 1) x

FIGURE 1.5 Graph of the function in Example 2.

x

x

3

4

Chapter 1: Functions

To find out, we could plot more points. But how would we then connect them? The basic question still remains: How do we know for sure what the graph looks like between the points we plot? Calculus answers this question, as we will see in Chapter 4. Meanwhile we will have to settle for plotting points and connecting them as best we can.

Representing a Function Numerically We have seen how a function may be represented algebraically by a formula (the area function) and visually by a graph (Example 2). Another way to represent a function is numerically, through a table of values. Numerical representations are often used by engineers and scientists. From an appropriate table of values, a graph of the function can be obtained using the method illustrated in Example 2, possibly with the aid of a computer. The graph consisting of only the points in the table is called a scatterplot.

EXAMPLE 3 Musical notes are pressure waves in the air. The data in Table 1.1 give recorded pressure displacement versus time in seconds of a musical note produced by a tuning fork. The table provides a representation of the pressure function over time. If we first make a scatterplot and then connect approximately the data points (t, p) from the table, we obtain the graph shown in Figure 1.6. p (pressure)

TABLE 1.1 Tuning fork data

Time

Pressure

Time

0.00091 0.00108 0.00125 0.00144 0.00162 0.00180 0.00198 0.00216 0.00234 0.00253 0.00271 0.00289 0.00307 0.00325 0.00344

-0.080 0.200 0.480 0.693 0.816 0.844 0.771 0.603 0.368 0.099 -0.141 -0.309 -0.348 -0.248 -0.041

0.00362 0.00379 0.00398 0.00416 0.00435 0.00453 0.00471 0.00489 0.00507 0.00525 0.00543 0.00562 0.00579 0.00598

Pressure 0.217 0.480 0.681 0.810 0.827 0.749 0.581 0.346 0.077 -0.164 -0.320 -0.354 -0.248 -0.035

1.0 0.8 0.6 0.4 0.2 –0.2 –0.4 –0.6

Data

0.001 0.002 0.003 0.004 0.005 0.006

t (sec)

FIGURE 1.6 A smooth curve through the plotted points gives a graph of the pressure function represented by Table 1.1 (Example 3).

The Vertical Line Test for a Function Not every curve in the coordinate plane can be the graph of a function. A function ƒ can have only one value ƒ(x) for each x in its domain, so no vertical line can intersect the graph of a function more than once. If a is in the domain of the function ƒ, then the vertical line x = a will intersect the graph of ƒ at the single point (a, ƒ(a)). A circle cannot be the graph of a function since some vertical lines intersect the circle twice. The circle in Figure 1.7a, however, does contain the graphs of two functions of x: the upper semicircle defined by the function ƒ(x) = 21 - x 2 and the lower semicircle defined by the function g (x) = - 21 - x 2 (Figures 1.7b and 1.7c). 4

5

1.1 Functions and Their Graphs y

y

y

–1 –1

0

1

x

–1

yx

2

0

1

2

x

3

x, -x,

x Ú 0 x 6 0,

whose graph is given in Figure 1.8. The right-hand side of the equation means that the function equals x if x Ú 0, and equals -x if x 6 0. Here are some other examples.

y y  f(x) 2 y1

1 –1

(c) y  –1  x 2

Sometimes a function is described by using different formulas on different parts of its domain. One example is the absolute value function ƒxƒ = e

FIGURE 1.8 The absolute value function has domain s - q , q d and range [0, q d .

–2

x

Piecewise-Defined Functions

1

y  –x

1 0

y  x

3

–1

x

FIGURE 1.7 (a) The circle is not the graph of a function; it fails the vertical line test. (b) The upper semicircle is the graph of a function ƒsxd = 21 - x 2 . (c) The lower semicircle is the graph of a function g sxd = - 21 - x 2 .

y

–3 –2

1

(b) y  1  x 2

(a) x 2  y 2  1

y  –x

0

0

yx

2

1

2

EXAMPLE 4

The function -x, ƒsxd = • x 2, 1,

x

FIGURE 1.9 To graph the function y = ƒsxd shown here, we apply different formulas to different parts of its domain (Example 4).

x 6 0 0 … x … 1 x 7 1

is defined on the entire real line but has values given by different formulas depending on the position of x. The values of ƒ are given by y = - x when x 6 0, y = x 2 when 0 … x … 1, and y = 1 when x 7 1. The function, however, is just one function whose domain is the entire set of real numbers (Figure 1.9).

y yx

3

The function whose value at any number x is the greatest integer less than or equal to x is called the greatest integer function or the integer floor function. It is denoted :x; . Figure 1.10 shows the graph. Observe that

2 y  ⎣x⎦

1 –2 –1

1

2

3

EXAMPLE 5

x

:2.4; = 2, :2; = 2,

:1.9; = 1, :0.2; = 0,

:0; = 0, : - 0.3; = - 1

: -1.2; = - 2, : -2; = - 2.

–2

FIGURE 1.10 The graph of the greatest integer function y = : x ; lies on or below the line y = x , so it provides an integer floor for x (Example 5).

EXAMPLE 6 The function whose value at any number x is the smallest integer greater than or equal to x is called the least integer function or the integer ceiling function. It is denoted < x = . Figure 1.11 shows the graph. For positive values of x, this function might represent, for example, the cost of parking x hours in a parking lot which charges $1 for each hour or part of an hour. 5

6

Chapter 1: Functions

Increasing and Decreasing Functions

y yx

3

If the graph of a function climbs or rises as you move from left to right, we say that the function is increasing. If the graph descends or falls as you move from left to right, the function is decreasing.

2 y  ⎡x⎤

1 –2 –1

1

2

x

3

DEFINITIONS Let ƒ be a function defined on an interval I and let x1 and x2 be any two points in I.

–1 –2

1. If ƒsx2) 7 ƒsx1 d whenever x1 6 x2 , then ƒ is said to be increasing on I. 2. If ƒsx2 d 6 ƒsx1 d whenever x1 6 x2 , then ƒ is said to be decreasing on I.

FIGURE 1.11 The graph of the least integer function y = < x = lies on or above the line y = x , so it provides an integer ceiling for x (Example 6).

It is important to realize that the definitions of increasing and decreasing functions must be satisfied for every pair of points x1 and x2 in I with x1 6 x2 . Because we use the inequality 6 to compare the function values, instead of … , it is sometimes said that ƒ is strictly increasing or decreasing on I. The interval I may be finite (also called bounded) or infinite (unbounded) and by definition never consists of a single point (Appendix 1).

EXAMPLE 7

The function graphed in Figure 1.9 is decreasing on s - q , 0] and increasing on [0, 1]. The function is neither increasing nor decreasing on the interval [1, q d because of the strict inequalities used to compare the function values in the definitions.

Even Functions and Odd Functions: Symmetry The graphs of even and odd functions have characteristic symmetry properties.

DEFINITIONS

A function y = ƒsxd is an even function of x if ƒs -xd = ƒsxd, odd function of x if ƒs -xd = - ƒsxd,

for every x in the function’s domain. y y  x2 (x, y)

(–x, y)

x

0 (a) y y  x3

0

(x, y) x

(–x, – y)

EXAMPLE 8 (b)

FIGURE 1.12 (a) The graph of y = x 2 (an even function) is symmetric about the y-axis. (b) The graph of y = x 3 (an odd function) is symmetric about the origin.

6

The names even and odd come from powers of x. If y is an even power of x, as in y = x 2 or y = x 4 , it is an even function of x because s -xd2 = x 2 and s - xd4 = x 4 . If y is an odd power of x, as in y = x or y = x 3 , it is an odd function of x because s -xd1 = - x and s -xd3 = - x 3 . The graph of an even function is symmetric about the y-axis. Since ƒs - xd = ƒsxd, a point (x, y) lies on the graph if and only if the point s -x, yd lies on the graph (Figure 1.12a). A reflection across the y-axis leaves the graph unchanged. The graph of an odd function is symmetric about the origin. Since ƒs -xd = - ƒsxd , a point (x, y) lies on the graph if and only if the point s -x, - yd lies on the graph (Figure 1.12b). Equivalently, a graph is symmetric about the origin if a rotation of 180° about the origin leaves the graph unchanged. Notice that the definitions imply that both x and -x must be in the domain of ƒ.

ƒsxd = x 2

Even function: s -xd2 = x 2 for all x; symmetry about y-axis.

ƒsxd = x 2 + 1

Even function: s - xd2 + 1 = x 2 + 1 for all x; symmetry about y-axis (Figure 1.13a).

ƒsxd = x

Odd function: s - xd = - x for all x; symmetry about the origin.

ƒsxd = x + 1

Not odd: ƒs -xd = - x + 1 , but -ƒsxd = - x - 1 . The two are not equal. Not even: s -xd + 1 Z x + 1 for all x Z 0 (Figure 1.13b).

1.1 Functions and Their Graphs

y

7

y

y  x2  1

yx1

y  x2

yx 1

1 x

0

–1

(a)

x

0

(b)

FIGURE 1.13 (a) When we add the constant term 1 to the function y = x 2 , the resulting function y = x 2 + 1 is still even and its graph is still symmetric about the y-axis. (b) When we add the constant term 1 to the function y = x , the resulting function y = x + 1 is no longer odd. The symmetry about the origin is lost (Example 8).

Common Functions A variety of important types of functions are frequently encountered in calculus. We identify and briefly describe them here. Linear Functions A function of the form ƒsxd = mx + b, for constants m and b, is called a linear function. Figure 1.14a shows an array of lines ƒsxd = mx where b = 0, so these lines pass through the origin. The function ƒsxd = x where m = 1 and b = 0 is called the identity function. Constant functions result when the slope m = 0 (Figure 1.14b). A linear function with positive slope whose graph passes through the origin is called a proportionality relationship.

m  –3

y m2 y  2x

y  –3x m  –1

m1

y

yx

m

y  –x

1 y x 2

0

x

1 2

2 1 0

(a)

y3 2

1

2

x

(b)

FIGURE 1.14 (a) Lines through the origin with slope m. (b) A constant function with slope m = 0.

DEFINITION Two variables y and x are proportional (to one another) if one is always a constant multiple of the other; that is, if y = kx for some nonzero constant k.

If the variable y is proportional to the reciprocal 1>x, then sometimes it is said that y is inversely proportional to x (because 1>x is the multiplicative inverse of x). Power Functions A function ƒsxd = x a , where a is a constant, is called a power function. There are several important cases to consider. 7

8

Chapter 1: Functions

(a) a = n, a positive integer. The graphs of ƒsxd = x n , for n = 1, 2, 3, 4, 5, are displayed in Figure 1.15. These functions are defined for all real values of x. Notice that as the power n gets larger, the curves tend to flatten toward the x-axis on the interval s -1, 1d, and also rise more steeply for ƒ x ƒ 7 1. Each curve passes through the point (1, 1) and through the origin. The graphs of functions with even powers are symmetric about the y-axis; those with odd powers are symmetric about the origin. The even-powered functions are decreasing on the interval s - q , 0] and increasing on [0, q d; the odd-powered functions are increasing over the entire real line s - q , q ). y

y

yx

1 –1

y

y  x2

1 0

–1

FIGURE 1.15

1

x

–1

y

y  x3

1

0

x

1

–1

–1

0

y y x5 

y  x4

1 x

1

–1

–1

1

0

1

x

–1

–1

0

1

x

–1

Graphs of ƒsxd = x n, n = 1, 2, 3, 4, 5, defined for - q 6 x 6 q .

(b) a = - 1

or

a = -2.

The graphs of the functions ƒsxd = x -1 = 1>x and gsxd = x -2 = 1>x 2 are shown in Figure 1.16. Both functions are defined for all x Z 0 (you can never divide by zero). The graph of y = 1>x is the hyperbola xy = 1, which approaches the coordinate axes far from the origin. The graph of y = 1>x 2 also approaches the coordinate axes. The graph of the function ƒ is symmetric about the origin; ƒ is decreasing on the intervals s - q , 0) and s0, q ). The graph of the function g is symmetric about the y-axis; g is increasing on s - q , 0) and decreasing on s0, q ). y y y  12 x

y  1x 1 0

1

x

Domain: x  0 Range: y  0

(a)

1 0

x 1 Domain: x  0 Range: y  0 (b)

FIGURE 1.16 Graphs of the power functions ƒsxd = x a for part (a) a = - 1 and for part (b) a = - 2.

(c) a =

2 1 1 3 , , , and . 2 3 2 3

3 x are the square root and cube The functions ƒsxd = x 1>2 = 2x and gsxd = x 1>3 = 2 root functions, respectively. The domain of the square root function is [0, q d, but the cube root function is defined for all real x. Their graphs are displayed in Figure 1.17 along with the graphs of y = x 3>2 and y = x 2>3 . (Recall that x 3>2 = sx 1>2 d3 and x 2>3 = sx 1>3 d2 .)

Polynomials

A function p is a polynomial if psxd = an x n + an - 1x n - 1 + Á + a1 x + a0

where n is a nonnegative integer and the numbers a0 , a1 , a2 , Á , an are real constants (called the coefficients of the polynomial). All polynomials have domain s - q , q d. If the 8

9

1.1 Functions and Their Graphs y y

y

y

yx

y  x

y  x 23

3

y  x

1 1 0

32

1 Domain: 0  x  Range: 0  y 

x

0

1 Domain: –  x  Range: –  y 

Graphs of the power functions ƒsxd = x a for a =

FIGURE 1.17

1

1 x

0

x

x

0 1 Domain: –  x  Range: 0  y 

1 Domain: 0  x  Range: 0  y 

2 1 1 3 , , , and . 2 3 2 3

leading coefficient an Z 0 and n 7 0, then n is called the degree of the polynomial. Linear functions with m Z 0 are polynomials of degree 1. Polynomials of degree 2, usually written as psxd = ax 2 + bx + c, are called quadratic functions. Likewise, cubic functions are polynomials psxd = ax 3 + bx 2 + cx + d of degree 3. Figure 1.18 shows the graphs of three polynomials. Techniques to graph polynomials are studied in Chapter 4. 3 2 y  x  x  2x  1 3 3 2 y

4

y y

2

–2

0

2

x

4

1

–2 –4 –6 –8 –10 –12

–2

–4 (a)

FIGURE 1.18



14x 3



9x 2

y  (x  2)4(x  1)3(x  1)

 11x  1

16

2 –1

–4

y

8x 4

x

2

–1

0

1

x

2

–16 (c)

(b)

Graphs of three polynomial functions.

Rational Functions A rational function is a quotient or ratio ƒ(x) = p(x)>q(x), where p and q are polynomials. The domain of a rational function is the set of all real x for which qsxd Z 0. The graphs of several rational functions are shown in Figure 1.19. y y

8 2 y  5x 2 8x  3 3x  2

y

4 2 2 y  2x  3 2 7x  4

–4

2

4

x

–5

5

0

2 10

–1

–2

y  11x3  2 2x  1

4 Line y  5 3

1

–2

6

x

–4

–2 0 –2

2

4

6

x

–4

–2

NOT TO SCALE

–4

–6 –8 (a)

FIGURE 1.19 of the graph.

(b)

(c)

Graphs of three rational functions. The straight red lines are called asymptotes and are not part

9

10

Chapter 1: Functions

Algebraic Functions Any function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots) lies within the class of algebraic functions. All rational functions are algebraic, but also included are more complicated functions (such as those satisfying an equation like y 3 - 9xy + x 3 = 0, studied in Section 3.7). Figure 1.20 displays the graphs of three algebraic functions. y  x 1/3(x  4)

y

y  x(1  x)2/5

y y  3 (x 2  1) 2/3 4 y

4 3 2 1

1

–1 –1 –2 –3

x

4

x

0

0

5 7

x

1

–1

(b)

(a)

FIGURE 1.20

(c)

Graphs of three algebraic functions.

Trigonometric Functions The six basic trigonometric functions are reviewed in Section 1.3. The graphs of the sine and cosine functions are shown in Figure 1.21. y

y

1

3 0

–



2

1

– 2 x

0 –1

–1

3 2

5 2

x

 2

(b) f (x)  cos x

(a) f (x)  sin x

FIGURE 1.21 Graphs of the sine and cosine functions.

Exponential Functions Functions of the form ƒsxd = a x , where the base a 7 0 is a positive constant and a Z 1, are called exponential functions. All exponential functions have domain s - q , q d and range s0, q d, so an exponential function never assumes the value 0. We discuss exponential functions in Section 1.5. The graphs of some exponential functions are shown in Figure 1.22. y

y y  10 x

y  10 –x

12

12

10

10

8

8 y

6 y  3x

4 2 –1

–0.5

FIGURE 1.22

10

0 (a)

1

6 4

y  2x 0.5

3 –x

2

y  2 –x x

Graphs of exponential functions.

–1

–0.5

0 (b)

0.5

1

x

11

1.1 Functions and Their Graphs

Logarithmic Functions These are the functions ƒsxd = loga x, where the base a Z 1 is a positive constant. They are the inverse functions of the exponential functions, and we discuss these functions in Section 1.6. Figure 1.23 shows the graphs of four logarithmic functions with various bases. In each case the domain is s0, q d and the range is s - q, q d. y

y

y  log 2 x y  log 3 x

1

0

1

–1

x y  log5 x

1

y  log10 x –1

FIGURE 1.23 Graphs of four logarithmic functions.

0

1

x

FIGURE 1.24 Graph of a catenary or hanging cable. (The Latin word catena means “chain.”)

Transcendental Functions These are functions that are not algebraic. They include the trigonometric, inverse trigonometric, exponential, and logarithmic functions, and many other functions as well. A particular example of a transcendental function is a catenary. Its graph has the shape of a cable, like a telephone line or electric cable, strung from one support to another and hanging freely under its own weight (Figure 1.24). The function defining the graph is discussed in Section 7.3.

Exercises 1.1 Functions In Exercises 1–6, find the domain and range of each function. 1. ƒsxd = 1 + x 2

2. ƒsxd = 1 - 2x

3. F(x) = 25x + 10 4 5. ƒstd = 3 - t

4. g(x) = 2x 2 - 3x 2 6. G(t) = 2 t - 16

In Exercises 7 and 8, which of the graphs are graphs of functions of x, and which are not? Give reasons for your answers. 7. a.

b.

y

y

8. a.

0

y

b.

x

0

x

Finding Formulas for Functions 9. Express the area and perimeter of an equilateral triangle as a function of the triangle’s side length x.

y

10. Express the side length of a square as a function of the length d of the square’s diagonal. Then express the area as a function of the diagonal length.

0

x

0

x

11. Express the edge length of a cube as a function of the cube’s diagonal length d. Then express the surface area and volume of the cube as a function of the diagonal length.

11

12

Chapter 1: Functions

12. A point P in the first quadrant lies on the graph of the function ƒsxd = 2x . Express the coordinates of P as functions of the slope of the line joining P to the origin.

y

31. a.

b. 2

13. Consider the point (x, y) lying on the graph of the line 2x + 4y = 5. Let L be the distance from the point (x, y) to the origin (0, 0). Write L as a function of x. 14. Consider the point (x, y) lying on the graph of y = 2x - 3. Let L be the distance between the points (x, y) and (4, 0). Write L as a function of y.

15. ƒsxd = 5 - 2x

16. ƒsxd = 1 - 2x - x 2

17. g sxd = 2ƒ x ƒ

18. g sxd = 2-x

19. F std = t> ƒ t ƒ

20. G std = 1> ƒ t ƒ

21. Find the domain of y =

x + 3 4 - 2x 2 - 9

(–2, –1) y

32. a.

y

b. (T, 1)

0 0

.

T 2

T

–A

x

T 2

T 3T 2T 2

t

The Greatest and Least Integer Functions 33. For what values of x is

x . x2 + 4

b. < x = = 0 ?

a. :x; = 0 ?

23. Graph the following equations and explain why they are not graphs of functions of x.

b. y 2 = x 2 a. ƒ y ƒ = x 24. Graph the following equations and explain why they are not graphs of functions of x. a. ƒ x ƒ + ƒ y ƒ = 1

x 1 (1, –1) (3, –1)

A

2

22. Find the range of y = 2 +

x

3

1

Functions and Graphs Find the domain and graph the functions in Exercises 15–20.

y

(–1, 1) (1, 1) 1

34. What real numbers x satisfy the equation :x ; = < x = ?

35. Does < - x = = - :x ; for all real x? Give reasons for your answer. 36. Graph the function

ƒsxd = e

b. ƒ x + y ƒ = 1

:x ;, x , x,

Increasing and Decreasing Functions Graph the functions in Exercises 37–46. What symmetries, if any, do the graphs have? Specify the intervals over which the function is increasing and the intervals where it is decreasing.

x … 1 x 7 1 x 6 0 0 … x

1 39. y = - x

40. y =

43. y = x 3>8

(1, 1)

45. y = - x

44. y = - 4 2x

3>2

2 y

30. a. 2

0

1

3

4

3

(2, 1) 5

2

x

1 –1

–1 –2 –3

12

2 y

b.

2

46. y = s - xd2>3

48. ƒsxd = x -5

47. ƒsxd = 3 0

1 ƒxƒ 42. y = 2 -x

Even and Odd Functions In Exercises 47–58, say whether the function is even, odd, or neither. Give reasons for your answer.

2

x

1 x2

y

b.

y 1

38. y = -

41. y = 2ƒ x ƒ

Find a formula for each function graphed in Exercises 29–32. 29. a.

37. y = - x 3

1

x 2 (2, –1)

t

2

49. ƒsxd = x + 1

50. ƒsxd = x 2 + x

51. gsxd = x 3 + x

52. gsxd = x 4 + 3x 2 - 1 x x - 1

53. gsxd =

1 x - 1

54. gsxd =

55. hstd =

1 t - 1

56. hstd = ƒ t 3 ƒ

2

57. hstd = 2t + 1

2

58. hstd = 2 ƒ t ƒ + 1

Theory and Examples 59. The variable s is proportional to t, and s = 25 when t = 75. Determine t when s = 60.

1.1 Functions and Their Graphs 60. Kinetic energy The kinetic energy K of a mass is proportional to the square of its velocity y. If K = 12,960 joules when y = 18 m>sec, what is K when y = 10 m>sec?

b. y = 5x

66. a. y = 5x

c. y = x 5

y g

61. The variables r and s are inversely proportional, and r = 6 when s = 4. Determine s when r = 10. 62. Boyle’s Law Boyle’s Law says that the volume V of a gas at constant temperature increases whenever the pressure P decreases, so that V and P are inversely proportional. If P = 14.7 lbs>in2 when V = 1000 in3, then what is V when P = 23.4 lbs>in2?

h

x x

14 x

x x

b. Confirm your findings in part (a) algebraically.

x

64. The accompanying figure shows a rectangle inscribed in an isosceles right triangle whose hypotenuse is 2 units long. a. Express the y-coordinate of P in terms of x. (You might start by writing an equation for the line AB.)

y

70. Three hundred books sell for $40 each, resulting in a revenue of (300)($40) = $12,000. For each $5 increase in the price, 25 fewer books are sold. Write the revenue R as a function of the number x of $5 increases.

B

P(x, ?)

A –1

0

x

1

x

In Exercises 65 and 66, match each equation with its graph. Do not use a graphing device, and give reasons for your answer. b. y = x 7

T 68. a. Graph the functions ƒsxd = 3>sx - 1d and g sxd = 2>sx + 1d together to identify the values of x for which 3 2 6 . x - 1 x + 1 b. Confirm your findings in part (a) algebraically. 69. For a curve to be symmetric about the x-axis, the point (x, y) must lie on the curve if and only if the point sx, -yd lies on the curve. Explain why a curve that is symmetric about the x-axis is not the graph of a function, unless the function is y = 0 .

b. Express the area of the rectangle in terms of x.

65. a. y = x 4

f

T 67. a. Graph the functions ƒsxd = x>2 and g sxd = 1 + s4>xd together to identify the values of x for which x 4 7 1 + x. 2

22 x

x

0

63. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 14 in. by 22 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.

x

13

71. A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs $5/ft for the legs and $10/ft for the hypotenuse, write the total cost C of construction as a function of h. 72. Industrial costs A power plant sits next to a river where the river is 800 ft wide. To lay a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs $180 per foot across the river and $100 per foot along the land.

c. y = x 10

2 mi P

x

Q

City

y 800 ft

g h

Power plant NOT TO SCALE

0 f

x

a. Suppose that the cable goes from the plant to a point Q on the opposite side that is x ft from the point P directly opposite the plant. Write a function C(x) that gives the cost of laying the cable in terms of the distance x. b. Generate a table of values to determine if the least expensive location for point Q is less than 2000 ft or greater than 2000 ft from point P.

13

14

Chapter 1: Functions

1.2

Combining Functions; Shifting and Scaling Graphs In this section we look at the main ways functions are combined or transformed to form new functions.

Sums, Differences, Products, and Quotients Like numbers, functions can be added, subtracted, multiplied, and divided (except where the denominator is zero) to produce new functions. If ƒ and g are functions, then for every x that belongs to the domains of both ƒ and g (that is, for x H Dsƒd ¨ Dsgd), we define functions ƒ + g, ƒ - g, and ƒg by the formulas s ƒ + gdsxd = ƒsxd + gsxd. sƒ - gdsxd = ƒsxd - gsxd. sƒgdsxd = ƒsxdgsxd. Notice that the + sign on the left-hand side of the first equation represents the operation of addition of functions, whereas the + on the right-hand side of the equation means addition of the real numbers ƒ(x) and g(x). At any point of Dsƒd ¨ Dsgd at which gsxd Z 0, we can also define the function ƒ>g by the formula ƒsxd ƒ a g b sxd = swhere gsxd Z 0d. gsxd Functions can also be multiplied by constants: If c is a real number, then the function cƒ is defined for all x in the domain of ƒ by scƒdsxd = cƒsxd.

EXAMPLE 1

The functions defined by the formulas ƒsxd = 2x

and

g sxd = 21 - x

have domains Dsƒd = [0, q d and Dsgd = s - q , 1]. The points common to these domains are the points [0, q d ¨ s - q , 1] = [0, 1]. The following table summarizes the formulas and domains for the various algebraic combinations of the two functions. We also write ƒ # g for the product function ƒg. Function

Formula

Domain

ƒ + g ƒ - g g - ƒ ƒ#g

s ƒ + gdsxd = 2x + 21 - x sƒ - gdsxd = 2x - 21 - x s g - ƒdsxd = 21 - x - 2x sƒ # gdsxd = ƒsxdgsxd = 2xs1 - xd ƒ ƒsxd x g sxd = gsxd = A 1 - x g gsxd 1 - x = sxd = ƒ ƒsxd A x

[0, 1] = Dsƒd ¨ Dsgd [0, 1] [0, 1] [0, 1]

ƒ>g g>ƒ

[0, 1) sx = 1 excludedd (0, 1] sx = 0 excludedd

The graph of the function ƒ + g is obtained from the graphs of ƒ and g by adding the corresponding y-coordinates ƒ(x) and g(x) at each point x H Dsƒd ¨ Dsgd, as in Figure 1.25. The graphs of ƒ + g and ƒ # g from Example 1 are shown in Figure 1.26. 14

15

1.2 Combining Functions; Shifting and Scaling Graphs

y

y

6

yfg

g(x)  1  x

8 y  ( f  g)(x)

1

y  g(x)

4 2

g(a)

y  f (x)

1 2

f(a)  g(a)

yf•g

f (a) x

a

0

f (x)  x

FIGURE 1.25 Graphical addition of two functions.

0

1 5

2 5

3 5

4 5

1

x

FIGURE 1.26 The domain of the function ƒ + g is the intersection of the domains of ƒ and g, the interval [0, 1] on the x-axis where these domains overlap. This interval is also the domain of the function ƒ # g (Example 1).

Composite Functions Composition is another method for combining functions.

DEFINITION If ƒ and g are functions, the composite function ƒ  g (“ƒ composed with g”) is defined by sƒ  gdsxd = ƒsgsxdd . The domain of ƒ  g consists of the numbers x in the domain of g for which g(x) lies in the domain of ƒ.

The definition implies that ƒ  g can be formed when the range of g lies in the domain of ƒ. To find sƒ  gdsxd, first find g(x) and second find ƒ(g(x)). Figure 1.27 pictures ƒ  g as a machine diagram and Figure 1.28 shows the composite as an arrow diagram. f g f(g(x)) x g x

g

g(x)

f

f

f (g(x))

FIGURE 1.27 Two functions can be composed at x whenever the value of one function at x lies in the domain of the other. The composite is denoted by ƒ  g.

g(x)

FIGURE 1.28 Arrow diagram for ƒ  g .

To evaluate the composite function g  ƒ (when defined), we find ƒ(x) first and then g(ƒ(x)). The domain of g  ƒ is the set of numbers x in the domain of ƒ such that ƒ(x) lies in the domain of g. The functions ƒ  g and g  ƒ are usually quite different. 15

16

Chapter 1: Functions

EXAMPLE 2

If ƒsxd = 2x and gsxd = x + 1, find

(a) sƒ  gdsxd

(b) sg  ƒdsxd

(c) sƒ  ƒdsxd

(d) sg  gdsxd .

Solution

Composite

Domain [ -1, q d

(a) sƒ  gdsxd = ƒsg sxdd = 2g sxd = 2x + 1 (b) sg  ƒdsxd = g sƒsxdd = ƒsxd + 1 = 2x + 1 (c) sƒ  ƒdsxd = ƒsƒsxdd = 2ƒsxd = 2 1x = x

1>4

(d) sg  gdsxd = g sg sxdd = g sxd + 1 = sx + 1d + 1 = x + 2

[0, q d [0, q d s - q, q d

To see why the domain of ƒ  g is [-1, q d, notice that gsxd = x + 1 is defined for all real x but belongs to the domain of ƒ only if x + 1 Ú 0, that is to say, when x Ú - 1. Notice that if ƒsxd = x 2 and gsxd = 2x, then sƒ  gdsxd = A 2x B 2 = x. However, the domain of ƒ  g is [0, q d, not s - q , q d, since 2x requires x Ú 0.

Shifting a Graph of a Function A common way to obtain a new function from an existing one is by adding a constant to each output of the existing function, or to its input variable. The graph of the new function is the graph of the original function shifted vertically or horizontally, as follows.

Shift Formulas Vertical Shifts

y = ƒsxd + k

y

y  x2  2 y  x2  1 y  x2

Shifts the graph of ƒ up k units if k 7 0 Shifts it down ƒ k ƒ units if k 6 0

Horizontal Shifts

y = ƒsx + hd

y  x2  2

Shifts the graph of ƒ left h units if h 7 0 Shifts it right ƒ h ƒ units if h 6 0

2 1 unit

EXAMPLE 3

1 –2

0 –1

2

x

2 units

–2

FIGURE 1.29 To shift the graph of ƒsxd = x 2 up (or down), we add positive (or negative) constants to the formula for ƒ (Examples 3a and b).

(a) Adding 1 to the right-hand side of the formula y = x 2 to get y = x 2 + 1 shifts the graph up 1 unit (Figure 1.29). (b) Adding -2 to the right-hand side of the formula y = x 2 to get y = x 2 - 2 shifts the graph down 2 units (Figure 1.29). (c) Adding 3 to x in y = x 2 to get y = sx + 3d2 shifts the graph 3 units to the left (Figure 1.30). (d) Adding -2 to x in y = ƒ x ƒ , and then adding -1 to the result, gives y = ƒ x - 2 ƒ - 1 and shifts the graph 2 units to the right and 1 unit down (Figure 1.31).

Scaling and Reflecting a Graph of a Function To scale the graph of a function y = ƒsxd is to stretch or compress it, vertically or horizontally. This is accomplished by multiplying the function ƒ, or the independent variable x, by an appropriate constant c. Reflections across the coordinate axes are special cases where c = - 1. 16

17

1.2 Combining Functions; Shifting and Scaling Graphs Add a positive constant to x.

Add a negative constant to x.

y

y y  x – 2 – 1

4 y  (x  3) 2

y  x2

y  (x  2) 2 1

1 –3

–4 1

0

–2

2

–1

4

6

x

x

2

FIGURE 1.30 To shift the graph of y = x 2 to the left, we add a positive constant to x (Example 3c). To shift the graph to the right, we add a negative constant to x.

FIGURE 1.31 Shifting the graph of y = ƒ x ƒ 2 units to the right and 1 unit down (Example 3d).

Vertical and Horizontal Scaling and Reflecting Formulas For c 7 1, the graph is scaled: y = cƒsxd Stretches the graph of ƒ vertically by a factor of c. 1 y = c ƒsxd

Compresses the graph of ƒ vertically by a factor of c.

y = ƒscxd Compresses the graph of ƒ horizontally by a factor of c. y = ƒsx>cd Stretches the graph of ƒ horizontally by a factor of c. For c = - 1, the graph is reflected: y = - ƒsxd Reflects the graph of ƒ across the x-axis. y = ƒs - xd Reflects the graph of ƒ across the y-axis.

EXAMPLE 4

Here we scale and reflect the graph of y = 2x.

(a) Vertical: Multiplying the right-hand side of y = 2x by 3 to get y = 32x stretches the graph vertically by a factor of 3, whereas multiplying by 1>3 compresses the graph by a factor of 3 (Figure 1.32). (b) Horizontal: The graph of y = 23x is a horizontal compression of the graph of y = 2x by a factor of 3, and y = 2x>3 is a horizontal stretching by a factor of 3 (Figure 1.33). Note that y = 23x = 232x so a horizontal compression may correspond to a vertical stretching by a different scaling factor. Likewise, a horizontal stretching may corr